There is a possible birth, although Pocklington isn't particularly near Leeds:
Jun 1883 Pocklington 9d 79
Caroline HAKNEY
The 1891 census has (indexed as ARNEY):
Bielby RG12/3901
William Akney, 46, ag lab, b Everingham
Mary A, 40
Agnes A, 16, dau
William A, 11, son
Margaret A, 9, dau
Alice A, 9, dau
Caroline A, 7, dau
Tom A, 4, son rest b Bielby
There is an Annie Akney, 20, b Bielby, (who the 1881 confirms is another daughter) who is a servant in Bramley, Leeds. She is still there in 1901 married to Albert E(dward) Turner.
The mother, Mary, is an imbecilic pauper inmate in Pocklington by 1901. I can't find Caroline at the moment.
Richard